How do I add the same Modifiers to all Linked Duplicates?
If you create a linked duplicate of an object (Alt+D) it will copy any existing modifiers. But the modifier stack of each duplicate can be individually manipulated without affecting the original. So if you retroactively add any modifiers to duplicates they will only be applied to one object.

You cannot add modifiers to multiple objects from the modifier panel. The modifier will only apply to the object selected last. However you can copy modifiers to a group selection, by selecting all the unmodified objects, and then the object with the modifiers, and use Ctrl+L to copy Modifiers.

Useful tip: to select all linked duplicates of a mesh use Shift+L or Select > Linked > "Object Data"





1st Image: Ctrl+L to copy modifiers from the last selected object

2nd Image: Shift+L to select all linked duplicates

S/O to @Ray Mairlot and @Gorgius for their comments, which I have redrafted as an answer for the sake of completion